Emergence, Locating and Measuring It - Systematic Design for Emergence in Cellular Nonlinear Networks - page 69

Fig. 4.11. Projection of the “2s5” CA family within the ( Tr,Clus,Var ) plane. Observe that
complex behaviors (gliders) are associated with either long transients or large variances.
Note the case of ID=440 which gave a large variance and a short transient (when simulation
time was set to T=400 but gave a large transient and a low variance when larger simulation
times were considered
Note that in the case of 2D neighborhoods, the variance range for “chaotic” pat-
terns is smaller, i.e. 0.02 < Var < 0.15.
Particularly interesting is to note that Var and Tr are complementary and or-
thogonal measures. In the case of Fig. 4.11, T=400 iterations were used in all runs.
For those cells where transient times were larger than the simulation times, the
variance is larger than the “chaotic” limit (in this case, Var >0.15) as seen for
ID=404. When the CA with ID=404 is simulated for 600 iterations, the long tran-
sient is clearly revealed. Though using large simulation times is computationally
costly.
Similar profiles can be easily drawn for other CA families. For instance, in
Fig. 4.12 a plot corresponding to the “1s7” family (with 16,384 members) is de-
picted.
4.5.1 Properties of Complexity Measures
The use of the Tr, Clus, and Var complexity measures, when applied to different
families of cellular automata reveal several interesting properties, as detailed next.
Let first discuss the Clus coefficient. As seen in Fig. 4.13, its distribution over the
entire family of semitotalistic CA is strongly influenced by the topology (in our
case, either 1D or 2D). While in the case of two-dimensional topologies (“2s9”
and “2s5” families) the plots reveal the existence of two CA categories, namely